The Fibonacci word is the limit of the sequence of words starting with "0" and satisfying rules $0 \to 01, 1 \to 0$. It's equivalent to have initial conditions $S_0 = 0, S_1 = 01$ and then recursion $S_n= S_{n-1}S_{n-2}$.

I want to know what words cannot appear as subwords in the limit $S_\infty$. At first I thought 000 and 11 were the only two that could not appear. Then I noticed 010101. Is there any characterization of which words can or cannot appear as subwords of the Fibonacci word?

Loosely related, this word appears as the cut sequence of the line of slope $\phi = (1 + \sqrt{5})/2$ though the origin.